I would like you to imagine that I am in the hospital right now and I go into the doctor's room for [STRIKE]the[/STRIKE] a/my check up. There are two doctors in [STRIKE]that[/STRIKE] the room. One of them is free and the other one is seeing a patient. The doctor who is free wants to [STRIKE]check[/STRIKE] examine me but I want to see the other doctor. [STRIKE]who is seeing the other patient at the moment. [/STRIKE] What would I say to [STRIKE]that[/STRIKE] the doctor who is free? Could I say "I want to see that doctor"?
After the check up, I come out and tell my mother [STRIKE]that[/STRIKE] that the second doctor checked me up or saw me.[STRIKE]"[/STRIKE] (There were no opening quotation marks so there was no need for closing quotation marks.)
Please check.
Note my corrections above.
There is no simple way of explaining what happened to your mother. You can't call the doctors "first" and "second", especially if your mother wasn't in the room so she wouldn't know what you meant by that.
You would have to explain it in full by saying something like "When I got into the room, there were two doctors but only one doctor was free to see me. However, I told him/her that I wanted to wait to be examined by the other doctor."
This wouldn't be a likely scenario in the UK. A consulting room usually has only one doctor. Medical appointments are confidential so there shouldn't be another patient in the room when you're being examined.